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Let $X,Y,Z$ be irreducible projective varieties over $\mathbb{C}$ (you can assume all of them are normal), and let $f:X\rightarrow Z, g: Y\rightarrow Z$ be two birational projective morphisms, such that

  • there is a common open set $U\subset Z$ with $f_U:f^{-1}(U)\simeq U$ and $g_U:g^{-1}(U)\simeq U$,
  • If $W:= Z-U$, then $f_W: f^{-1}(W)\rightarrow W$ and $g_W: g^{-1}(W)\rightarrow W$ are projective bundles of ranks $m,n$ respectively, and
  • codim$(W)=1+m+n$.

Is it true that $X\times_{Z}Y$ is integral? I have seen a statement along this line, but can't prove it.

Any help would be really appreciated.

Edit: Thanks to R. van Dobben de Bruyn's comment, the answer to the question as I stated earlier was false. I had skipped the codimension condition.

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    $\begingroup$ This is false. For example, let $f$ and $g$ both be the blowup $\tilde X \to X$ at a point $p$ in a smooth variety $X$ of dimension $d \geq 3$. Then the exceptional divisor is $\mathbf P^{d-1}$, so the fibre product has a subvariety isomorphic to $\mathbf P^{d-1} \times \mathbf P^{d-1}$ lying over $p$. But $2d-2 > d$, so $\tilde X \times_X \tilde X$ has components of different dimensions! $\endgroup$ Jul 17, 2020 at 14:24
  • $\begingroup$ @R.vanDobbendeBruyn : Thanks for the answer! Actually, there were more conditions on the codimension of $W$, but I thought that it was not required. I have edited my question. $\endgroup$ Jul 17, 2020 at 16:54

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